Also practice the same thing with the left hand, and then with both hands together.

The do the same thing in other keys. E.g. in D major use this pattern:

D F# A D' F#' A# D'' ...

and so on for other key signatures.

I am not sure I see any great advantage in practicing the other sequences you mention, such as D, F, A.

It would be useful to get a book of scales and arpeggios such as the ones published by the exam boards (e.g. ABRSM) as they give the sort of scales and arpeggios that are encountered most frequently.

A couple of advantages of doing arpeggios like this would be: (a) this is the way arpeggios frequently arise in music by Mozart, Beethoven, Schubert etc. and (b) the "thumb under" technique need to play the arpeggios fluidly this way encourages a flexible and fluid way of playing.

Most of what I have just said is pretty much in line with my experience of classical piano, and may not be quite the same as other genres. Jazz pianists may see things very differently, I guess!

Agree with looking at the exam syllabi, but ABRSM advocates starting arpeggios with black key start notes to start with another digit than the thumb (1), so thumb can be used on subsequent white key - like Eb major for example.
– TimDec 30 '17 at 12:39

Yes, I wasn't advocating the fingering 1-2-3-1-2-3-5 for all keys. For some (e.g. inversions), it is better to use 1-2-4-1-2-4-5, and if starting with the a black note, better to avoid starting with the thumb. Hence my recommendation to get a copy of a book of scales and arpeggios.
– Old JohnDec 30 '17 at 12:47

More important than the notes and fingering is ergonomically HOW to play. The arm is a machine designed to obey certain laws of physics. Although, we can and do bend those rules but bending the rules of physics comes with a price: Mediocrity or injury.

Each finger is a different length, they have different types of tendons and bones, they radiate at different angles, several different muscles can move them and many of these muscles are very weak and prone to fatigue. Ironically, those are the ones most pianists use and think they have to build strength and endurance in order to play. They actually need to not use those muscles and use the proper big ones. BTW, there are no muscles in your fingers.

It would be best that you find a teacher who will not allow you to touch the piano until you master the movements properly and understand the physics of the piano and your arm. The second you first touch a piano your brain maps the movement and if you strike the key using incorrect movement, the brain wires it in. If there is a leap or scale that eludes you, it isn't because you need more practice, it is because an errant movement is getting in your way and always will. Practicing more won't fix anything. Practicing properly will. You need a teacher to tell you how to move based upon physics, not errant movement that their teacher taught them and their's before them. It isn't fact because that is how they were taught. Even "facts" can be wrong.

Playing arpeggios involves a lot of movement from up/down, in/out, moving the arm, using pronator and supinator muscles, adjusting the elbow, adjusting the forearm, playing with gravity, forward shifts, grouping, then they all need to be combined. There are just as many things NOT to do such as pressing, abducting, crossing the thumb under the palm, isolating fingers, ulnar and radial deviation, trying to play with a still and quiet hand, trying to equalize the fingers . . . The arm plays the fingers, the fingers don't play the piano.

The point is, your first teacher and first movements must be perfect for, once a bad habit is set in stone, it can take a lifetime to eradicate.

Some teachers like to let the student "figure it out" which some students do as they improve and experiment. But it would be better to teach them everything properly before they start playing. We don't let people drive cars or fly planes that way. Why pianists? Oh, is it because nobody dies? What is a career doomed for mediocrity?

Most teachers should not be teaching. Maybe if we put them all on a plane and said "Land it."

I mainly do this in order to get some memorization of the chords of
each scale:

Then do it! there is nothing bad about it, also the progression makes sense in a way that the VII degree resolves back to I. (It contains the same notes as the V7 - without root).

what is the most common or effective way of practicing arpeggios on a
piano.

There is many ways of practicing arpeggios on the piano and many of them make sense for what they are gaining for. Your suggested way has one big disadvantage, that is that it is pretty time consuming since you are talking about 84 chords (not counting inversions) while many of them are repetitions. Repetition is one of the main aspects of practice, but if you want to move a bit quicker through the circle of fifth I would recommend the following:

Start in C major then move downwards, like this:

C |
Bm7♭5 |
Am |

Then (in your head already in G major), move a step down:

G |
F♯m7♭5 |
Em |

and so on...

Put together you play I, VII, VI and change the key up a fifth, by moving one step further down (You might want to change to an octave higher at some point). You then will end up with 36 different chords.

Inversions of course are a good thing to practice, ABRSM-tests often ask for inversions when asking for arpeggios.

You can then also add an octave and practice the chords like this, also giving a different shape for the inversions:

RH: c-1 e-2 g-3 c-5
LH: c-5 e-4 g-2 c-1

(NOTE: that the fingering is not consistent when moving through different chords with mixed black and white keys)